Question

A class consisting of 10 students takes a test. The scores are approximately normally distributed with a mean of 74 and a standard deviation of 11. The teacher wants to curve the grades in the following manner: the top 5% get As, the next highest 12% get Bs, the bottom 5% get Fs, and the next lowest 12% get Ds. The rest will get Cs. Find the Z-scores for each line and the test score corresponding to each z score

Answer 1

Solution :

Given that,

mean = = 74

standard deviation = = 11

n = 10

=74

₌ / n = 11 10 = 3.4785

Using standard normal table,

P( Z < z) = 5%
P(Z < z) = 0.05

z =-1.64

Using z-score formula,

= z * +

= -1.64 * 3.4785 + 74

= 67.7926

=67.79

P( Z > z) = 5%

P(Z > z) = 0.05

1 - P( Z < z) = 0.05

P(Z < z) = 1 - 0.05

P(Z < z) = 0.95

z = 1.64

Using z-score formula,

= z * +

= 1.64 * 3.4785 + 74

= 79.7047

=79.70

P( Z < z) = 12%
P(Z < z) = 0.12

z = -1.17

Using z-score formula,

= z * +

= -1.17 * 3.4785 + 74

= 69.9301

=69.93

P( Z > z) = 12%

P(Z > z) = 0.12

1 - P( Z < z) = 0.12

P(Z < z) = 1 - 0.12

P(Z < z) = 0.88

z = 1.17

Using z-score formula,

= z * +

= -1.64 * 3.4785 + 74

= 78.0698

=78.07